Galileo found more difficulty than one should expect in his endeavours to obtain an exact measure of the power of gravity ; and indeed could not obtain one that was satisfac tory. But the difficulty of the task, and his struggle to ac complish it, were big with advantages to science. A body falls so fast, that a considerable error in the conclusion arises from a very small error in estimating the time ; and the great difficulty was how to estimate the time. It was in this casting about for a measure of a small por tion of time that Galileo first thought of the pendulum. His penetrating and sagacious mind enabled him to see that there must be a fixed proportion between the time of a vibration and that of falling through its length, although his mathematical knowledge did not yet enable him to find it out ; he saw an immediate consequence of this, if true, namely, that the vibrations of two pendulums should be in the subduplicate ratio of the lengths, because this must be the proportions of the times of falling through those lengths. This he would try ; and he found that it was so. Delighted with this success, he immediately compared the time of falling from the top of the great dome with that of a pendulous vibration, by making a pendulum of such a length that it performed precisely one vibration in the time of the fall. In this time the body, moving with the final velocity, would describe a space double of .hat fallen through. He then counted with patience the number of vibrations made by his pendulum in an in terval of time, measured by the transit of two stars. Thus lie obtained the time, and the velocity generated in that time, by the uniform action of gravity. Galileo made this to be about 31 feet of our measure in a second, and said that it was certainly somewhat more ; because his expe riments on falling bodies convinced him that their motion is retarded by the air.
These efforts and resources of an ingenious mind are worthy of record, and are instructive to others. But Ga lileo did not attain the accuracy in this measure that we now possess. The honour of the accurate statement of the time of a pendulous oscillation, and that of the fall through its length, was reserved for Mr Huygens. This proportion was determined by him by a most ingenious and elegant physico-mathematical process. He also gave us the pendulum clock, by which time can be measured with as much accuracy as a line can be divided.
Aided by these inventions, we have now obtained the most precise measure of the accelerating powers of gravi ty; and we can now say that its intensity is such in the lati tude of London, that by acting uniformly on a body for one second of time, it generates in it the velocity of 32 feet two inches per second, and a heavy body falls 16 feet one inch in that time.
These are standard numbers, of continual use in all me chanical discussions, and should be carefully kept in re membrance. Not only so, but we should acquire distinct notions of them in this respect, viz. as standard numbers. Gravity is known to us in two ways ; our most familiar acquaintance with it is as a pressure, which we feel when we carry a heavy body. With this we can compare the
pressure of a spring, the exertion of an animal, the pres sure of a stream of water or wind, the intensity of an at traction, &c. by setting them in opposition and equilibrium. The philosopher, and especially the physical astronomer, and cultivator of the Newtonian philosophy, is well ac quainted with gravity as an accelerating and a moving force, capable of accelerating, retarding, or deflecting the body-in which it inheres, or on whose intimate particles it acts without intermedium. He can compare the gravity of a stone with that of the moon, or of Jupiter, or with the force that produces the precession of the equinoxes. The general mechanician, observing that all other pressures, such as that of a spring, of an animal, &c. are also moving forces, by combining those two aspects of gravity, makes a most important use of it, by comparing other forces with weights, and thence inferring the motions which those forces will produce. Thus, knowing that an arrow oz. weight, by falling 18 inches, acquires the velocity of feet per second, he infers, that when drawn to the head by a bow of 62 pounds, it will be discharged with the velocity of 233 feet per second.
We shall therefore, in future, compare every force with gravity, and express the accelerating power of this stan dard by 32, meaning, that by acting on every particle or a body for a second, it will generate the velocity of N. feet per second, and cause the body to describe 16 feet with a motion uniformly accelerated. We may find it convenient, on some occasions, to use the numbers 386, and 193, which are the inches in and 16A- feet.
The questions that interest us at present are, those con cerning the relations between the time t of any fall, the height h of that fall, and the velocity v that is uniformly acquired in falling ; so that when any one of those things is given, the others may be found out.
I. Since the variations of velocity are proportionalao the limes in which they are produced, we have : t"=32 : 32r and v'=32‘" and t".— N. B. The time t is always supposed to be a number of seconds, and the height h a number of feet, and the veloci ty -v a number of feet uniformly moved over in one second.
A falling body, therefore, acquires an increment of 32 feet per second in every second of its fall, and an ascending body has its velocity lessened as much (luring every second of its rise. A body falling during four seconds, acquires the velocity of 128 feet per second.
But if the body has been projected downward with the velocity of 100 feet per second, then at the end of 4", it is moving at the rate of 228 feet per second.
A body projected straight upwards with the velocity of 160 feet per second, will at the end of the first second of its rise, have the velocity 128. At the end of 2" it will be moving at the rate of 96 feet per second. Its velocity at the end of the third second will be 64. At the end of the fourth second, it will be 32, and at the end of five seconds it will stop, and begin to fall.